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Rangeazimuth decouple beamforming for frequency diverse array with Costassequence modulated frequency offsets
EURASIP Journal on Advances in Signal Processing volume 2016, Article number: 124 (2016)
Abstract
Different from the phasedarray using the same carrier frequency for each transmit element, the frequency diverse array (FDA) uses a small frequency offset across the array elements to produce rangeangledependent transmit beampattern. FDA radar provides new application capabilities and potentials due to its rangedependent transmit array beampattern, but the FDA using linearly increasing frequency offsets will produce a range and angle coupled transmit beampattern. In order to decouple the rangeazimuth beampattern for FDA radar, this paper proposes a uniform linear array (ULA) FDA using Costassequence modulated frequency offsets to produce randomlike energy distribution in the transmit beampattern and thumbtack transmitreceive beampattern. In doing so, the range and angle of targets can be unambiguously estimated through matched filtering and subspace decomposition algorithms in the receiver signal processor. Moreover, randomlike energy distributed beampattern can also be utilized for low probability of intercept (LPI) radar applications. Numerical results show that the proposed scheme outperforms the standard FDA in focusing the transmit energy, especially in the range dimension.
Introduction
Active phasedarray has been widely adopted in many applications such as radar, electronic warfare, radio astronomy, etc., because it can steer the beam electronically with high effectiveness [1]. The offered directional gain is useful for detecting/tracking weak targets and suppressing sidelobe interferences in other directions [2]. Since phasedarray has rangeindependent transmit beampattern, the range and angle of targets cannot be unambiguously estimated from the beamforming output peaks. If we want to steer the array beams to multiple different range cells, multiple antennas or a multibeam antenna will be required. More importantly, controlling rangedependent energy distribution becomes an increasingly important requirement in many applications. While techniques exist in mitigating rangedependent interferences, e.g., spacetime adaptive processing [3], they generally require a high computational cost.
In order to overcome the disadvantage of phasedarray radar, the frequency diverse array (FDA) radar using a small frequency offset across the array elements was proposed in 2006 [4]. This steppedfrequency offset results in that the beam can scan the space in a periodic manner [5, 6]. Its beamforming focusing direction will change as a function of the range, angle, time, and even the frequency offset [7]. These characteristics contrast with the rangeindependent transmit beampattern in a phasearray radar. FDA radar is different from orthogonal frequency division multiplexing (OFDM) radar [8] and multipleinput multipleoutput (MIMO) radar [9, 10]. OFDM radar uses orthogonal subcarriers, but nonorthogonal carriers are employed in FDA radar. MIMO radar aims to provide noncoherent waveforms to obtain increased degrees of freedom (DOFs), whereas FDA radar transmits overlapping signals with closely spaced frequencies to provide additional functionality. FDA radar is also different from conventional frequency scanning radar using the frequency increments as a function of time for all the elements [11], but FDA frequency offsets are characterized by the element index [12]. Another similar concept is the timemodulation array [13], which weights each element using on/off switching operation. FDA was investigated in [14] as a rangedependent beam with applications in suppressing range ambiguous clutter. Secmen et al. [5] described the time and angle periodicity of FDA radiation pattern. Higgins and Blunt [15] explored rangeangle coupled beamforming in FDA. Additional studies to exploit FDA rangedependent beampattern characteristics were reported in [16, 17]. In fact, with the introduction of frequency increment, the FDA apparent angle will be different from its nominal beam scanning angle.
Due to its promising application potentials [18], FDA has sparked many interesting investigations [19–21]. Since FDA offers a rangeangledependent beampattern, it is of great importance as this provides a potential for rangeangle localization of targets, but the transmit beampattern of a standard FDA using linearly increasing frequency offsets is coupled in the rangeangle dimension. This limits its application for unambiguously estimating target parameters. To decouple the rangeangle coupling response of targets, a simple rangeazimuth localization of targets is proposed in [22] by adopting a uniform linear array (ULA) doublepulse FDA radar. This doublepulse FDA radar transmits two pulses with zero and nonzero frequency offsets, respectively. In [23], a subarraybased FDA is proposed for target rangeangle estimation. Furthermore, a transmit subaperturing is designed in [20] with convex optimization, so that the range and angle responses are decoupled and the equivalent transmit beam can be focused in a certain rangeangle sector to localize the targets.
In [19], a nonuniform linear array is adopted for the FDA. However, the transmitter and receiver must be placed accurately. Another nonuniform linear array for FDA is attempted to suppress/locate rangedependent interference/target in [12], but the carrier frequency and/or frequency increments cannot be altered in realtime because it requires relocating the elements mechanically. Logarithmically increasing frequency offsets [24] or timedependent frequency offsets [25] are also suggested to decouple the rangeangle beampattern, but they result in poor beamforming performance, especially in the range dimension. In fact, the best decoupling approach is to form a dotshaped beampattern rather than an “S”shaped beampattern. Such a dotshaped rangeangle beampattern is synthesized in [26] by a symmetrical FDA using multicarrier frequency offsets and convex optimization, and in [27] by the use of random frequency offsets. Nevertheless, these two schemes are very difficult to implement in practical array systems.
In this paper, we analyze the reason of FDA range and angle coupled transmit beampattern and thus propose the rangeazimuth decouple beamforming for FDA radar using Costassequence modulated frequency offsets. The rest of this paper is organized as follows. Section 2 provides a brief introduction to basic FDA radar scheme and motivation of this paper. Then, Section 3 proposes the rangeazimuth decoupling beamforming for the FDA using Costassequence modulated frequency offsets. Finally, numerical results are provided in Section 4 and concluding summaries are drawn in Section 5.
Basic FDA radar and motivation
In conventional phasedarrays, it is assumed that the same waveform is radiated by each array elements. Different from conventional phasedarrays, FDA elements can be excited by either the same waveform or different waveforms. Without loss of generality, we assume that the waveform radiated from each array element is identical with a frequency offset of Δ f Hz, as shown in Fig. 1, where the xaxis and xaxis are defined as along the linear array arrangement and the zeroazimuth angle, respectively. Therefore, the monochromatic continuous signal transmitted by the mth element is
where w _{ m } is the amplitude weighting and w _{ e }(f _{ m }) is the element pattern factor which can be expressed as [28]
where θ is the direction angle, c _{0} is the speed of light, and sinc(x)= sin(x)/x. In order to evaluate the impacts of the frequency offsets on the w _{ e }(f _{ m }), we define the following performance metric:
Suppose Δ f=5 kHz, Fig. 2 shows the ratio \(\frac {w_{e}(f_{m})}{w_{e}(f_{0})}\) as a function of the number of array elements. It is noticed that the frequency offsets have ignorable effects on the amplitude changes of the w _{ e }(f _{ m }), and thus, w _{ e }(f _{ m })=1 is adopted in subsequent discussions.
Different from traditional phasedarray antenna, which uses the same carrier frequency for all the array elements, the FDA uses different carrier frequencies with small frequency offsets for the array elements. This implies that the FDA elements will have frequencydependent beam steering component. The radiation frequency f _{ m } is [4]
with f _{0} and M being the carrier frequency and number of array elements, respectively. The signal arriving at a farfield point with slant range r for the first element and direction angle θ can be expressed as
where r _{ m } can be approximated by
with d being the element spacing.
To avoid aliasing effects, the element spacing should be smaller than the half wavelength of the highest frequency signal. Generally, positive frequency offset Δ f is assumed in most of literatures. Certainly, negative frequency offset Δ f is also feasible for the FDA radar. Therefore, in this paper the uniform array element spacing is designed as
If the amplitude and beam steering components are all equal to one, namely, w _{ m }=1 and w _{ e }(f _{ m })=1, the array factor seen at the target position (r,θ) can be derived as [19]
where the common phase Φ _{0} is \(\Phi _{0}=2\pi f_{0}\left (t\frac {r}{c_{0}}\right)\).
To get a closedform expression, we approximate the phase term m ^{2} Δ f d sinθ/c _{0} as m Δ f d sinθ/c _{0}. To make the assumption reasonable, an empirical requirement can be employed:
Suppose the carrier frequency is f _{0}=10 GHz, the azimuth angle is θ=π/3 and the element spacing is half of the maximal wavelength. Figure 3 shows the maximum allowable frequency increment as a function of the number of array elements, M. Since Eq. (9) is just an empirical phase requirement, the high offsets shown in Fig. 2 cannot be used in practice due to bandwidth constraints. In order to avoid also frequency decorrelation to happen in target response, in most literature the frequency offset Δ f is assumed to be smaller than one thousandth of the transmitted signal bandwidth B _{ r }. It is observed that the assumption requirement is easily achieved for practical FDA radar systems. In this case, Eq. (8) can be rewritten in a closedform as
where Φ _{1} is
According to Eq. (10), the transmit beampattern achieves the maximum at
When the time variable t is fixed, the FDA beampattern will be coupled in the range and angle dimensions, caused by the synchronous linearly changing between frequency increment and element spacing. If this synchronization is damaged, the FDA may yield uncoupled rangeangle beampattern. Hence, we have two potential solutions:

1. Linearly increasing frequency offsets and nonuniform linear array

2. Uniform linear array and nonlinearly increasing frequency offsets
In this paper, we use Costassequence modulated frequency increments, namely, the second case.
FDA radar using Costassequence modulated frequency offsets
In the Costas frequency coding scheme, the columns represent M contiguous time slices (each of duration t _{ b }) and the rows represent M distinct frequencies, equally spaced by Δ f _{0}. We use this scheme for the FDA, namely, only one carrier frequency is transmitted by any one of he M FDA elements and each carrier frequency is used only once. The construction algorithms for Costas signals were discussed by Golomb and Taylor [29]. The coding sequence, the order of used frequencies is a concise way to describe the coding matrix. With regard to the difference matrix, note that the top row and the leftmost column are headings and not part of the matrix. The element of the difference matrix in row i and column j is
where a _{ i } is the ith element of the coding sequence. The remaining locations (where i+j>M) are left blank. This equation implies that the first row is formed by taking differences between adjacent elements in the coding sequence, the second row is formed by taking differences between nextadjacent elements, and so on.
Construction of Costas codes can be understood as a construction of steppedfrequency waveforms, where the pulse width τ is divided into M subpulses, each of width τ _{1}. Within each group of M subpulses, the frequency is increased by Δ f from one subpulse to the next. The total signal bandwidth is (M−1)Δ f and there are M subpulses; each subpulse has the duration of 1/Δ f, then the timebandwidth product of the transmitted signal is (M−1)Δ f×M×1/Δ f=(M−1)M. Costas codes are similar to steppedfrequency waveforms, except that the frequencies for the subpulse are selected in a random fashion, according to some predetermined rule or logic. Figure 4 compares the hopping orders of LFM and Costas coding schemes, where the xaxis and yaxis denote the time and frequency, respectively.
The normalized complex envelope of the Costas signal can be expressed as [30]
where u _{ m }(t)= exp(j2π f _{ m } t), 0≤t≤τ _{1}. Note that the subpulses are separated in timedomain. It is easily understood that the hopping order strongly affects the ambiguity function of the signal. The ambiguity function can be predicted roughly by overlapping a copy of the binary matrix on itself and then shifting one relative to the other according to the desired delay and Doppler. The corresponding ambiguity function of the matched filter is
where
As noted in Eq. (14), in the standard Costas sequence the subpulses corresponding to each element are not aligned. However, in the FDA the transmitted signals from all the elements should be aligned in timedomain; otherwise, the beampattern of the whole array will be decided mainly by some particular elements for a given instant. To avoid this problem, we use the Costassequence modulated frequency offsets in a timealigned way. Taking the Costas sequence illustrated in Fig. 4 as an example, the adopted frequency indexes in the seven time intervals are {4,7,1,6,5,2,3}. Accordingly, our method allows for a sevenelement FDA and the seven elements use the frequency offsets 4Δ f _{0}, 7Δ f _{0}, 1Δ f _{0}, 6Δ f _{0}, 5Δ f _{0}, 2Δ f _{0}, and 3Δ f _{0} with Δ f _{0} being the hopping frequency step, respectively. That is, similar to conventional FDA, all the signals are transmitted simultaneously from the Costas modulated FDA.
The frequency fed to the mth element of the FDA using Costassequence modulated frequency offsets can be generally written as
where Δ f _{ m } is the frequency increment for the mth element. The transmitted signal of the mth element can then be expressed as
For an ideal point target at the range r and azimuth θ, the received echo corresponding to the mth antenna is
By demodulating the received returns with the transmit signal, we can get the baseband signal:
For notation convenience, the above equation can be simply rewritten as
where ϕ _{ m } is
In the single snapshot case, the received noisefree echo of one ideal target can be represented as the following receive steering vector
where ^{T} is the transpose operator. For the multitarget case, the received echo vector for the kth snapshot can then be expressed as
where α _{ p }(k), r _{ p }, and θ _{ p } are the reflection coefficient, slant range, and azimuth angle for the pth target at the kth snapshot, respectively, P is the target number, K is the snapshot number, and n(k) is the M×1 additive receiver noise vector. Note that the target reflection coefficient α _{ p }(k) may vary from shapshot to snapshot [31].
Adaptive beamforming algorithms can be used to optimally design the weighting vector w to synthesize the desired transmitreceive beampattern. Specifically, when the nonadaptive beamforming algorithm is adopted, the weighting vector is
where θ _{0} and r _{0} denote the angle and range of the desired target, respectively. In this case, the maximum is steered to the expected location (r _{0},θ _{0}). The FDA radar transmitreceive beampattern can then be expressed as
It is noticed that, like a phasedarray radar, the FDA radar has coherent transmit processing gain; however, the FDA radar directional gain depends on both the range and angle parameters, whereas the phasedarray radar directional gain depends only on the range parameter. This rangeangledependent beam provides a potential approach to suppress rangedependent interferences and noise.
The conventional nonadaptive beamforming is known to be optimal in the sense that it provides the highest possible output signaltonoise ratio (SNR) and signaltointerference plus noise ratio (SINR) in the background of white Gaussian noise [32]. The output SINR of the FDA radar can be evaluated by
where \({\sigma ^{2}_{s}}\) is the variance of the desired target signal, \({\sigma ^{2}_{i}}\) is the variance of the ith interference, and \({\sigma ^{2}_{n}}\) is the noise variance. If the target is observed in the background of few weak interferences which are well separated from the target, the interferencetonoise power can be attributed to the noise term only. In this case, the SINR for the FDA radar simplifies to
which means that the FDA radar has an equivalent robustness again noise.
In contrast, if the target is observed in the background of strong interferences, then we can fairly consider the noise power to be negligible as compared to the interference power. In such a case, we have
The FDA using Costassequence modulated frequency offsets will make the transmitreceive beampattern mainlobe approximate an ideal thumbtack response, as provided in the next section. In this case, the SINR (Eq. (31)) can be simplified as
where \(\overline {\sigma }^{2}_{i}\) denotes the mean of \({\sigma ^{2}_{i}}\). Thus, it is expected that this FDA radar has better robustness against interferences than both conventional phasedarray radar and standard FDA radar.
Numerical results
It is well known that when a given delayDoppler shift results in a coincidence of N points, the ambiguity function is expected to yield a peak of approximately N/M at the corresponding delayDoppler coordinate. For the linearly frequency modulated (LFM) coding case, only delay and Doppler shifts of equal number of units, namely, τ=m t _{ b },ν=m Δ f,m=0,±1,…,±(M−1), will cause an overlap of dots, and the number of coinciding dots will be N=M−m. This hints at a diagonal ridge in the ambiguity function, along the line ν=Δ f τ/t _{ b }. What is unique for a Costas signal is that the number of coinciding dots cannot be larger than one for all but the zeroshift case, where all dots coincide (N=M). This property implies a narrow peak of the ambiguity function at the origin and low sidelobes elsewhere. If Δ f=1/τ _{1}, the exact ambiguity function values at the grid points will be either 1 or 0, according to the corresponding number of coinciding dots.
As an example, we consider the WelchConstructed Costas coding sequence for M=12, namely, the chosen positions are 1,2,4,8,3,6,12,11,9,5,10, and 7. This implies that the frequency offsets are not linearly increasing as the element index, but change as the Costas sequence, namely, {1Δ f _{0},2Δ f _{0},4Δ f _{0},8Δ f _{0},3Δ f _{0},6Δ f _{0}, 12Δ f _{0},11Δ f _{0},9Δ f _{0},5Δ f _{0},10Δ f _{0},7Δ f _{0}}. Suppose Δ f _{0}=100 k H z, Fig. 5 a shows the ambiguity function for a monochromatic waveform with Costassequence modulated frequency offsets for the FDA radar. As a comparison, suppose the following parameters for the LFM waveform: bandwidth is 10 MHz, starting frequency is 0 Hz and pulse duration is 1 μ s. Figure 5 b, c show the ambiguity function for a LFM waveform with Costassequence modulated frequency offsets for FDA and phasedarray radars, respectively. It is noticed that the use of Costassequence modulated frequency offsets produces a more focused peak. The ambiguity function sidelobes are usually lower than 20 log10(2/M), but the near sidelobes are higher decaying in a manner typical of the sidelobes of a signal with a rectangular spectrum.
Next, we consider the transmit beampattern for the arrays. Suppose Δ f=5 kHz and use also the WelchConstructed Costas sequence with M=12, Fig. 6 a shows the Costassequence modulated FDA using uniform transmit weighting coefficients. It is seen that the transmit beampattern has randomlike peak distribution without obvious peaks. This implies that it is difficult to be detected or localized by an unfriendly detector without knowing the specified coding sequence for the frequency increments, and it can be exploited to develop low probability interception (LPI) or radio frequency (RF) stealth radar techniques [33]. However, with the knowledge of the specific Costas sequences used in the FDA transmitter, we can recover the thumbtack transmitreceive beampattern for the targets. As a comparison, Fig. 6 b provides the standard FDA using also uniform transmit weighting coefficients, which produces rangeangledependent and rangeangle coupling transmit beampattern. This implies that it can be exploited for rangeangle localization of targets, but it may produce ambiguous results due to the rangeangle coupling beampattern. Differently, the traditional phasedarray generate angledependent only transmit beampattern, as shown in Fig. 6 c, but it provides no range information of the targets because of its rangeindependent beampattern [34].
Finally, we consider another example to compare the differences between different length of Costas sequences. Using the WelchConstructed Costas coding sequence with M=18, namely, the chosen positions are 1,2,4,8,16,13,7,14,9,18,17,15,11,3,6,12,5, and 10. Figure 7 shows the corresponding transmit beampattern and radar ambiguity function for Δ f _{0}=100 kHz. The results validate again that the FDA using Costassequence modulated frequency offsets produces randomlike peak distribution without obvious peaks, and consequently, it is difficult to be detected by unfriendly detectors. Nevertheless, it generates also a focused peak in the radar ambiguity function. Therefore, we can conclude that LPI can be achieved by the FDA radar using Costassequence modulated frequency offsets.
Conclusions
This paper proposed a ULA FDA using Costas sequence modulated frequency offsets to decouple the rangeangledependent beampattern by producing randomlike energy distribution in the transmit beampattern. Moreover, a thumbtack transmitreceive beampattern can be obtained at the receiver. In doing so, the range and angle of targets can be solely estimated through matched filtering and subspace decomposition algorithms in the receiver signal processor. Numerical results show that the proposed scheme outperforms the standard FDA in focusing the transmit energy, especially in the range dimension. This Costassequence modulated FDA radar can be exploited for LPI radar applications due to the random transmit beampattern and thumbtack transmitreceive beampattern. Such a topic is planned for our future work.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under grant 61571081, Sichuan Province Science Fund for Distinguished Young Scholars under grant 2013JQ0003, and Sichuan Technology Research and Development fund under grant 2015GZ0211.
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The authors declare that they have no competing interests.
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Wang, Z., Wang, W. & Shao, H. Rangeazimuth decouple beamforming for frequency diverse array with Costassequence modulated frequency offsets. EURASIP J. Adv. Signal Process. 2016, 124 (2016). https://doi.org/10.1186/s1363401604223
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Keywords
 Frequency diverse array (FDA) radar
 Frequency offset
 Rangedependent
 Decouple
 Costas sequence